Counting Factorizations

4000ms

262144K

Description:

The prime factorization of a positive integer $$$m$$$ is the unique way to write it as $$$\displaystyle m=p_1^{e_1}\cdot p_2^{e_2}\cdot \ldots \cdot p_k^{e_k}$$$, where $$$p_1, p_2, \ldots, p_k$$$ are prime numbers, $$$p_1 < p_2 < \ldots < p_k$$$ and $$$e_1, e_2, \ldots, e_k$$$ are positive integers.

For each positive integer $$$m$$$, $$$f(m)$$$ is defined as the multiset of all numbers in its prime factorization, that is $$$f(m)=\{p_1,e_1,p_2,e_2,\ldots,p_k,e_k\}$$$.

For example, $$$f(24)=\{2,3,3,1\}$$$, $$$f(5)=\{1,5\}$$$ and $$$f(1)=\{\}$$$.

You are given a list consisting of $$$2n$$$ integers $$$a_1, a_2, \ldots, a_{2n}$$$. Count how many positive integers $$$m$$$ satisfy that $$$f(m)=\{a_1, a_2, \ldots, a_{2n}\}$$$. Since this value may be large, print it modulo $$$998\,244\,353$$$.

Input:

The first line contains one integer $$$n$$$ ($$$1\le n \le 2022$$$).

The second line contains $$$2n$$$ integers $$$a_1, a_2, \ldots, a_{2n}$$$ ($$$1\le a_i\le 10^6$$$) — the given list.

Output:

Print one integer, the number of positive integers $$$m$$$ satisfying $$$f(m)=\{a_1, a_2, \ldots, a_{2n}\}$$$ modulo $$$998\,244\,353$$$.

Sample Input:

2
1 3 2 3

Sample Output:

Sample Input:

2
2 2 3 5

Sample Output:

Sample Input:

1
1 4

Sample Output:

Note:

In the first sample, the two values of $$$m$$$ such that $$$f(m)=\{1,2,3,3\}$$$ are $$$m=24$$$ and $$$m=54$$$. Their prime factorizations are $$$24=2^3\cdot 3^1$$$ and $$$54=2^1\cdot 3^3$$$.

In the second sample, the five values of $$$m$$$ such that $$$f(m)=\{2,2,3,5\}$$$ are $$$200, 225, 288, 500$$$ and $$$972$$$.

In the third sample, there is no value of $$$m$$$ such that $$$f(m)=\{1,4\}$$$. Neither $$$1^4$$$ nor $$$4^1$$$ are prime factorizations because $$$1$$$ and $$$4$$$ are not primes.

Informação

Provedor Codeforces

Código CF1794D